# MIT open course Physics 1 - Mechanics by Walter Lewin - Lecture 28 Hydrostatics, Archimedes' Principle, and Fluid Dynamics

Материал готовится,

пожалуйста, возвращайтесь позднее

пожалуйста, возвращайтесь позднее

Today we're going to continue with playing with liquids.

If I have an object that floats, a simple cylinder that floats in some liquid, the area is A here, the mass of the cylinder is M.

The density of the cylinder is rho and its length is l and the surface area is A.

So this is l.

And let the liquid line be here, and the fluid has a density rho fluid.

I call this level y1, this level y2.

The separation is h, and right on top here, there is the atmospheric pressure P2, which is the same as it is here on the liquid.

And here we have a pressure P1 in the liquid.

For this object to float we need equilibrium between, on the one hand, the force Mg and the buoyant force.

There is a force up here which I call F1, and there is a force down here which I call F2 —

barometric pressure.

The force is always perpendicular to the surface.

There couldn't be any tangential component because then the air starts to flow, and it's static.

And here we have F1, which contains the hydrostatic pressure.

So P1 minus P2 —

as we learned last time from Pascal —

equals rho of the fluid g to the minus y2 minus y1, which is h.

So that's the difference between the pressure P1 and P2.

For this to be in equilibrium, F1 minus F2 minus Mg has to be zero, and this we call the buoyant force.

And "buoyant" is spelt in a very strange way: b-u-o-y-a-n-t.

I always have to think about that.

It's the buoyant force.

F1 equals the area times P1 and F2 is the area times P2, so it is the area times P1 minus P2, and that is rho fluids times g times h.

And when you look at this, this is exactly the weight of the displaced fluid.

The area times h is the volume of the fluid which is displaced by this cylinder, and you multiply it by its density, that gives it mass.

Multiply it by g, that gives it weight.

So this is the weight of the displaced fluids.

And this is a very special case of a general principle which is called Archimedes' principle.

Archimedes' principle is as follows: The buoyant force on an immersed body has the same magnitude as the weight of the fluid which is displaced by the body.

According to legend Archimedes thought about this while he was taking a bath, and I have a picture of that here —

I don't know from when that dates —

but you see him there in his bath, but what you also see are there are two crowns.

And there is a reason why those crowns are there.

Archimedes lived in the third century B.C.

Archimedes had been given the task to determine whether a crown that was made for King Hieron II was pure gold.

The problem for him was to determine the density of this crown —

which is a very irregular-shaped object —

without destroying it.

And the legend has it that as Archimedes was taking a bath, he found the solution.

He rushed naked through the streets of Syracuse and he shouted, "Eureka! Eureka! Eureka!" which means, "I found it! I found it!" What did he find? What did he think of? He had the great vision to do the following: You take the crown and you weigh it in a normal way.

So the weight of the crown —

I call it W1 —

is the volume of the crown times the density of which it is made.

If it is gold, it should be 19.3, I believe, and so this is the mass of the crown and this is the weight of the crown.

Now he takes the crown and he immerses it in water.

And he has a spring balance, and he weighs it again.

If I have an object that floats, a simple cylinder that floats in some liquid, the area is A here, the mass of the cylinder is M.

The density of the cylinder is rho and its length is l and the surface area is A.

So this is l.

And let the liquid line be here, and the fluid has a density rho fluid.

I call this level y1, this level y2.

The separation is h, and right on top here, there is the atmospheric pressure P2, which is the same as it is here on the liquid.

And here we have a pressure P1 in the liquid.

For this object to float we need equilibrium between, on the one hand, the force Mg and the buoyant force.

There is a force up here which I call F1, and there is a force down here which I call F2 —

barometric pressure.

The force is always perpendicular to the surface.

There couldn't be any tangential component because then the air starts to flow, and it's static.

And here we have F1, which contains the hydrostatic pressure.

So P1 minus P2 —

as we learned last time from Pascal —

equals rho of the fluid g to the minus y2 minus y1, which is h.

So that's the difference between the pressure P1 and P2.

For this to be in equilibrium, F1 minus F2 minus Mg has to be zero, and this we call the buoyant force.

And "buoyant" is spelt in a very strange way: b-u-o-y-a-n-t.

I always have to think about that.

It's the buoyant force.

F1 equals the area times P1 and F2 is the area times P2, so it is the area times P1 minus P2, and that is rho fluids times g times h.

And when you look at this, this is exactly the weight of the displaced fluid.

The area times h is the volume of the fluid which is displaced by this cylinder, and you multiply it by its density, that gives it mass.

Multiply it by g, that gives it weight.

So this is the weight of the displaced fluids.

And this is a very special case of a general principle which is called Archimedes' principle.

Archimedes' principle is as follows: The buoyant force on an immersed body has the same magnitude as the weight of the fluid which is displaced by the body.

According to legend Archimedes thought about this while he was taking a bath, and I have a picture of that here —

I don't know from when that dates —

but you see him there in his bath, but what you also see are there are two crowns.

And there is a reason why those crowns are there.

Archimedes lived in the third century B.C.

Archimedes had been given the task to determine whether a crown that was made for King Hieron II was pure gold.

The problem for him was to determine the density of this crown —

which is a very irregular-shaped object —

without destroying it.

And the legend has it that as Archimedes was taking a bath, he found the solution.

He rushed naked through the streets of Syracuse and he shouted, "Eureka! Eureka! Eureka!" which means, "I found it! I found it!" What did he find? What did he think of? He had the great vision to do the following: You take the crown and you weigh it in a normal way.

So the weight of the crown —

I call it W1 —

is the volume of the crown times the density of which it is made.

If it is gold, it should be 19.3, I believe, and so this is the mass of the crown and this is the weight of the crown.

Now he takes the crown and he immerses it in water.

And he has a spring balance, and he weighs it again.

Загрузка...

Выбрать следующее задание

Ты добавил

Выбрать следующее задание

Ты добавил